Negative binomial distribution
The negative binomial distribution ( Pascal distribution) is a discrete probability distribution, one of the three Panjer distributions.
Describes the number of attempts required to obtain a given number of successes in a Bernoulli process.
In addition to the Poisson distribution, the negative binomial distribution is the most important number of claims distribution in insurance mathematics. There it is particularly used as a damage index distribution in health insurance, on rare automobile liability or comprehensive insurance.
- 2.1 Expectation value
- 2.2 variance
- 2.3 Coefficient of variation
- 2.4 skewness
- 2.5 Characteristic Function
- 2.6 Generating function
- 2.7 moment generating function
- 3.1 Relationship to the geometric distribution
- 4.1 When the umpteenth Skat Paula wins for the 10th time?
Derivation of the negative binomial distribution
One can describe this distribution with replacement using the urn model: In an urn there are two kinds of balls ( dichotomous population ). The proportion of the balls of first grade. The probability that a ball is drawn first grade, is so.
It is now so long pulled a ball and put back until the first result just balls of first grade. It is a random variable: define " number of attempts until the first time resulting successes ". The number of experiments is the quantity. has countably infinitely many possible forms.
The probability that attempts were needed to achieve success, therefore, is calculated according to the following consideration:
It should have already taken place at the present time trials. There were a total of drawn balls of first grade. The probability is determined by the binomial distribution of the random variables: specified " number of balls of first grade for tests ":
The probability that now another ball of first grade is drawn, then
A random variable is called so negative binomial distribution with the parameters (number of successful attempts ) and ( likelihood of success in a single trial ) when for them the probability function
Can be entered.
Alternative definition
A discrete random variable subject to the negative binomial distribution with parameters and, if the probabilities
For has.
Both definitions are into relationship; So while the first definition asks for the number of attempts ( successful and unsuccessful ) to the occurrence of the -th success, the alternative representation for the number of failures until the occurrence of the -th success interested. The successes are not counted. Then the random variable referred to only the number of unsuccessful attempts.
Properties of the negative binomial distribution
- A special case for the negative binomial distribution, the geometric distribution. Here are more interested in the number of failures until the first success occurs.
- The negative binomial distributions belong to the Panjer class.
- The sum of mutually independent geometrically distributed random variables with the same parameter is negative - binomial distribution with parameters and.
Expected value
The expected value is determined as
In the alternative definition of the expected value is to be smaller, ie.
Variance
The variance of the negative binomial distribution is given for both definitions by
The variance is always greater than the expected value ( overdispersion ) in the alternative definition.
Coefficient of variation
From expectation and variance immediately gives the coefficient of variation to
Skew
The skewness is given by:
Characteristic function
The characteristic feature is in the form
Generating function
For the generating function is obtained
Moment generating function
The moment generating function of the negative binomial distribution is
Relations with other distributions
Relationship to the geometric distribution
The negative binomial distribution is for the geometric distribution over. On the other hand, the sum of identical, independent, geometrically distributed random variables with the same parameters, a negative binomial random variable with parameters and.
Examples
When umpteenth Skat Paula wins for the 10th time?
The student Paula plays tonight Skat. From long experience, she knows that she wins every 5th game. Winning is defined as follows: You must first get through a game stimuli, then they must win this game.
As they at eight clock has statistics lecture tomorrow, should not be too long of evening. That's why she decided to go after the 10th match won home. Suppose that a game lasts ( a generous estimate ), about 4 minutes. What is the probability they can go home after two hours, so after 30 games?
We go with our considerations analogous prior to the above:
What is the probability it has won 9 times in 29 games? We calculate this probability with the binomial distribution, in terms of the urn model with 29 trials and 9 balls of first grade:
The probability of making the 10th win at the 30th game, is now
This probability now seems to be very small. The graph of the negative binomial distribution X shows that overall, the chances remain very small. As there is poor Paula ever come to bed? We can reassure them: It is enough even to ask how many attempts Paula at most needs, it need not even be exactly 30.
The probability that a maximum of 30 trials are needed, the distribution function F ( x ) of the negative binomial distribution at x = 30, which here the sum of the probabilities P (X = 0 ) P ( x = 1) P (X = 2) ... P ( X = 30 ) results. A look at the graph of the distribution function shows that if Paula with a 50% probability is satisfied, they would have to complete a maximum of 50 games, which would be 50.4 min = 200 min = 3h 20 min. To get a 80 % chance of their 10 wins, they would have no more than about 70 games, so just under 5 hours. Perhaps Paula should change their strategy but the number of games.
Discrete univariate distributions for finite sets: Benford | Bernoulli | beta - binomial | binomial | categorical | hypergeometric | Rademacher | generalized binomial | Zipf | Zipf - Mandelbrot
Discrete univariate distributions for infinite sets: Boltzmann | Conway - Maxwell - Poisson | negative binomial | extended negative binomial | Compound Poisson | discrete uniform | discrete phase -type | Gauss - Kuzmin | geometric | logarithmic | parabolic fractal | Poisson | Poisson - Gamma | Skellam | Yule- Simon | Zeta
Continuous univariate distributions with compact interval: Beta | Cantor | Kumaraswamy | raised cosine | triangle | U - square | steady uniform | Wigner semicircle
Continuous univariate distributions with half-open interval: Beta prime | Bose -Einstein | Burr | Chi-Square | Coxian | Erlang | Exponential | F | Fermi -Dirac | Folded normal | Fréchet | Gamma | Gamma Gamma | extreme | generalized inverse Gaussian | semi logistically | semi- normal | Hotelling's T-square | hyper- exponential | hypoexponential | inverse chi-square | scale - inverse- chi-square | inverse Normal | inverse gamma | Levy | log-normal | log- logistically | Maxwell -Boltzmann | Maxwell speed | Nakagami | not centered chi-square | Pareto | Phase -Type | Rayleigh | relativistic Breit-Wigner | Rice | Rosin -Rammler | shifted Gompertz | truncated normal | Type -2 Gumbel | Weibull | Wilks ' lambda
Continuous univariate distributions with unbounded interval: Cauchy | extreme | exponentially Power | Fishers z | Fisher - Tippett ( Gumbel ) | generalized hyperbolic | Hyperbolic- secant | Landau | Laplace | alpha- stable | logistics | normal ( Gaussian ) | normal - inverse Gauß'sch | skew - normal | Student's t | Type -1 Gumbel | Variance gamma | Voigt
Discrete multivariate distributions: Ewen | multinomial | Dirichlet compound multinomial
Continuous multivariate distributions: Dirichlet | generalized Dirichlet | multivariate normal | multivariate Student | normal scaled inverse gamma | Normal - Gamma
Multivariate matrix distributions: Inverse Wishart | matrix normal | Wishart
- Probability distribution